It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.

中文翻译：

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变分泊松同调

众所周知，如果两个哈密顿算子之一具有相应的一阶变分泊松同调的精确信息，就可以确定所谓的双哈密顿PDE可积性的Lenard-Magri方案的有效性。在本文的第一部分中，我们解释了如何通过使用相应的通用李超代数或李保形来引入各种同调复合物，包括李超代数和泊松同调复合物，以及基本和简化的李保形代数与泊松顶点代数同调复合物。超代数。最相关的是基本和简化的Poisson顶点代数同调复合物的某些子复合物，我们将其（广义地）与广义de Rham复合物和广义变分复合物一起识别。在本文的第二部分中，我们计算了广义de Rham复数的同调，并且，通过对长精确序列的详细研究，我们计算了具有可逆前导系数的任何拟常数系数哈密顿算子的广义变分复数的同调。对于后者，我们使用附录中开发的一些微分线性代数。